There are a variety of hand-held air pumps for bicycle. These air pumps are provided with a single air chamber, dual (multiple) air chambers, a single stroke, a dual stroke, etc. These air pumps share one thing in common in that they are all provided with a check valve to ensure the nonreturn of the air being pumped into a bicycle tire from the air pumps. The location and the structure of the check valve are various, depending on the overall design of the air pumps. For example, the U.S. Pat. No. 5,165,876 discloses an air pump comprising a piston rod 39 which is provided at the tail end thereof with a non-return valve 37 and a non-return valve cap 38. The current market situation of the bicycle air pump is chaotic in that the air valve connection heads are often independently designed and manufactured for a specific air pump, and that the air valve connection heads are therefore incompatible with the air pumps other than the specific air pump. Such a chaotic situation as described above is wasteful and uneconomical.